Many future robot systems may be required to operate in environments that are highly unstructured with varying dynamical properties, and are active, i.e., possessing means of self-actuation. The development of control technologies for unpredictable environments is a critical first step in efforts to design autonomous robotic systems. Specifically, algorithms devised for such a purpose must exhibit (i) robustness to parametric uncertainties in dynamic models and (ii) the ability to adapt rapidly enough to parametric variations in order to insure operational performance. Although a significant volume of literature may be found on the problems of robust control and adaptive control, many issues pertinent to robust and adaptive control algorithms for large magnitude and high frequency parametric uncertainties remain unresolved.
While positive displacement (PD) control schemes with gravity compensation have been shown to be globally stable for setpoint control, they cannot guarantee stability in the presence of parametric uncertainties.
Proportion, integral and differential (PID) control with gravity compensation is globally stable for setpoint control, even in the presence of uncertainty, but the resulting closed loop system is rather sluggish. This could potentially generate large transient errors, causing tool and robot damage during contact operations. Compliance and stiffness control schemes guarantee local stability and function adequately in quasi-static contact situations with well characterized stationary environments.
Impedance control has been applied to a wide spectrum of contact applications. [N. Hogan, "Impedance control: An approach to manipulation: Part I--Theory, Part II--Implementation, Part III--Applications," Journal of Dynamic Systems, Measurement and Control, 1985, pp. 1-24] The implementations are, however, limited to static environments. A detailed stability analysis of impedance control is available in N. Hogan, "On the stability of manipulators performing contact tasks," IEEE Journal of Robotics and Automation, Vol. 4, No. 6, December 1988.
Many robust control schemes have been proposed and implemented for pure motion and compliant motion control of robots and have demonstrated the ability to stabilize closed loop behavior in the presence of bounded parametric uncertainties. In addition, recent progress in adaptive control has enabled on-line accommodation of unknown robot mass and inertial parameters and payloads. [J. J. Slotine, W. Li, "Adaptive manipulator control: A case study," IEEE Transactions on Automatic Control, Vol. 33, No. 11, November 1988, pp. 995-1003]
There are a number of applications in which the foregoing state-of-the-art control technologies would fall short of prescribed system performance. Consider two scenarios relevant to space applications: (1) robot-assisted extravehicular activity and (2) autonomous fresh sample acquisition during precursor science [S. T. Venkataraman, N. Marzwell, "Telerobots for robust space operations," (to appear) Proceedings SPACE 92, Denver, Colo., 1992].
In the first scenario, voice-activated robots would assist astronauts by fetching and returning tools, helping transport large and heavy payloads, etc. In the second scenario, robotic elements interact with soil and rock media to extract core samples for scientific analysis. Note, however, that in both scenarios, the environment dynamics is partially or completely unknown. In addition, since astronauts are active, i.e., capable of self-actuation, and environments are unpredictable, e.g., rocks could contain unknown crevices, shear planes and cracks, the rate at which the robot control law accommodates the uncertainties must be explicitly controllable. Thus, the robot control law needs to be adjusted according to the rate of change in the environment. The following table summarizes the environmental effects in the two problem areas.
______________________________________ Application Unmodeled Phenomena Effect on Dynamics ______________________________________ Robotic Cracks in Rocks Unstructured Dynamics Coring Holes/Vesicles in Rocks High Frequency Pressure Variation in Variations Regolith High Amplitude Variations Astronaut Unpredictable Human Unstructured Muscular Assistant Movements Dynamics Varying Muscle Stiffness Unpredictable Muscular Actuation ______________________________________
Further, in both of the above applications, the actual dynamic characteristics of the environments are highly nonlinear. For example, research in biomechanical systems [J. M. Winters, L. Stark, A-H. Seif-Naraghi, "An analysis of the sources of musculoskeletal system impedance," Journal of Biomechanics, Vol. 21, No. 12, 1988, pp. 1011-1025; J. M. Winters, L. Stark, "Estimated mechanical properties of synergistic muscles involved in movements of a variety of human joints," Journal of Biomechanics, Vol. 21, No. 12, 1988, pp. 1027-1041], has indicated that the linear spring, dashpot model used in equation (8) of A. V. Hill, "The heat of shortening and the dynamic constants of muscle," Proceedings Royal Society, Vol. 126B, pp. 136-195, would be inadequate.
Models consisting of exponential and higher order polynomial terms have been suggested for the viscoelastic properties of the muscles, and their contractile dynamics have been modeled using first-order ordinary differential equation [Winters and Stark, supra]. Further, the strong interdependences between a human neuronal impedance (input impedance of the human muscular structure), his mechanical impedance [Winters, Stark and Seif-Naraghi, supra], coupled with the variations caused by fatigue [M. Pousson, J. Van Hoecke, F. Goubel, "Changes in elastic characteristics of human muscle induced by eccentric exercise," Journal of Biomechanics, Vol. 23, No. 4, 1990, pp. 343-348] cause the environmental models to be nonautonomous.
Similarly, environmental characteristics of rocks and regolith are extremely complex. For example, the models suggested in D. S. Rowley, F. C. Appl, "Analysis of Surface Set Diamond Bit Performance," Society of Petroleum Engineers Journal, September 1969, pp. 301-310, for diamond coring suggests a nonlinear dependence of the normal thrust force on the rock hardness, characteristics of the diamond matrix, drill diameters and the drilling rate. In addition, the relationship is nonautonomous due to the effect of temperature on drill characteristics, diamond wear and chip removal.
State-of-the-art PD control cannot guarantee system stability during the above mentioned robotic tasks. The sluggishness in system response with PID control could result in tool damage during sampling tasks and cause excessive human fatigue during astronaut assistance. The effectiveness of conventional compliance and stiffness control methods during autonomous sampling and astronaut assistance operations would be extremely limited given the complex nonautonomous nature of environment dynamics.
A primary concern with sliding mode robust control approaches is the large switching gains required and the constant chattering around the sliding surface. During autonomous sampling, this typically causes excessive tool wear, sample degradation and actuator saturation. Control switching, even as a phenomenon, cannot be recommended for man-machine systems. Although chattering can be potentially eliminated by the use of interpolation manifolds [Asada, et al., supra, pp. 139-157], it is not recommended for applications referred to in the table above since their size cannot be determined accurately apriori.
The present invention focuses on two key control requirements, control convergence and control robustness. When perfect model information is available, the closed loop convergence must be controllable depending, of course, upon environmental characteristics. For example, during astronaut assistance, robotic tasks must be completed in some specified time interval. The latter property has been referred to as finite time control systems in the control literature.
The second issue pertains to the development of robust control laws that do not require high frequency control switches. With this motivation, a theoretical framework that allows terminal control convergence is developed, wherein the convergence time is finite and controllable. A terminal sliding mode robust control law is proposed to deal with model uncertainties. It is shown that the proposed method leads to greater guaranteed precision in all control cases discussed herein.
The concept of applying sliding modes to control emerged from earlier work on variable structure systems, notably the work of A. F. Fillipov, "Differential equations with discontinuous right-hand sides," Annals Mathematical Society Transactions, Vol. 42, 1964. In principle, it revolves around the choice of a control law that forces the closed loop system behavior to be identical to a sliding surface. Typically, the closed loop system dynamics represents controlled system error behavior. As a result, one can model closed loop behavior through an appropriate choice of sliding surfaces. If a sliding surface is chosen such that s=e+.lambda.e=0, where e is the trajectory error and .lambda. is a positive constant chosen by designer, then exponential error convergence occurs. Consider, for example, the system EQU x=.function.(x)+u (1)
where x is the system state, .function. is a smooth function of x, and u is the control. A control law of the form u=u.sub.o =-.function.(x)+X.sub.d -.lambda.e will result in s=0. If the initial condition, s(t.sub.o), is zero then the system converges exponentially. When s(t.sub.o).noteq.0, a control switch of the form K Sgn(s)K&gt;0 may be added to u.sub.o, where K is a constant, and Sgn(s) is 1, 0, and -1 if s is greater than zero, equal to zero, or less than zero (i.e., negative), respectively, to force the system to converge towards the sliding surface. At s=0, the switch is deactivated, and the previous arguments apply. It can be shown that the system will reach the sliding surface in finite time [Asada, et al., supra].
In the presence of parametric uncertainties where only .function.(x), an estimate of .function.(x), is available, control law takes the following form: ##EQU5## which yields s=.function.(x)-.function.(x)-K Sgn(s). By choosing K&gt;.parallel..function.(x)-.function.(x).parallel., Lyapunov stability and convergence towards the sliding surface ,s, can be ensured. [Asada, et al., supra]. The closed loop system does not actually stay on the sliding surface, since at s=0, s.noteq.0 it chatters in the neighborhood of the sliding surface [Asada, et al., supra]. Conventional sliding mode control, therefore, guarantees exponential stability with full model information and asymptotic stability in the presence of uncertainties. The design of interpolation regions is typically performed off-line using bounds on uncertainty and the expected system response in the neighborhood of the sliding surface.
The concept of terminally sliding surfaces may be developed from first principles and applied to control synthesis for nonlinear systems. The performance thus obtained can be compared to those with the conventional sliding mode control. To enhance convergence properties of dynamical systems, the concept of terminal attractors was introduced in M. Zak, "Terminal attractors for content addressable memory in neural networks," Physics Letters, Vol. 133, pp. 218-222, 1988. Since then terminally sliding surfaces have demonstrated considerable success in neural learning [J. Bahren and S. Gulati, "Self-organizing neuromorphic architecture for manipulator inverse kinematics," NATO ASI Series, Ed. C. S. G. Lee, Vol. 44, 1990]. It has the basic form of a cubic parabola: EQU x=-x.sup.1/3 ( 3)
with an equilibrium point at x=0. Integrating between t.sub.initial and t.sub.equilibrium, ##EQU6## This implies that Equation (4) settles into equilibrium in finite time. For an additional discussion of terminal sliders in coping with variations in unstructured environments, see S. T. Venkataraman and S. Gulati, "Terminal Sliding Modes: A New Approach to Nonlinear Control Synthesis," Fifth International Conference on Advanced Robotics, Pisa, Italy, Vol. 1, pp. 443-448, Jun. 19-22, 1992, which by this reference is hereby made a part hereof. This property has also been applied for finite time control of distributed parameter systems. [S. Jayasuriya and A. R. Diaz, "Performance enhancement of distributed parameter systems by a class of nonlinear control," Proceedings Conference on Decision and Control, Los Angeles, Calif., December 1987, pp. 2125-2126]. The better convergence results from increased local stability. A detailed discussion on terminal attractor may be found in Zak, supra.
The most general form for a first-order terminal attractor would be x+X(x)=0, where X is bounded for a bounded x Sgn(X)=Sgn x and ##EQU7## Such systems are Lipschitzian in any error, .epsilon., neighborhood of the equilibrium point, but are non-Lipschitzian at the equilibrium point itself. An intuitive argument about the dynamic behavior of such systems is provided in M. Zak, "Cumulative effect at the soil surface due to shear wave propagation," ASME Journal of Applied Mechanics, Vol. 50, 1983, pp. 227-229. To analyze such systems using Lyapunov methods, postulate the following: Given a dynamical system of the following form x+X(x)=0, and a Lyaponov function candidate V(x), where V is bounded for founded x.parallel.V(x.noteq.0).parallel.&gt;0,.parallel.V(x=0).parallel.=0, if EQU V+g(V)=0 (5)
where g is a smooth function of V, such that V has the terminal attractor property described above, the dynamical system is terminally stable. For example, the system ##EQU8## would be terminally stable since ##EQU9## implies ##EQU10## In this invention, terminal attractors of the form ##EQU11## are exploited, where .alpha.&gt;0 and .beta..sub.n,.beta..sub.d =(2i+1), where i belongs to the set of selected positive integers I so that both .beta..sub.n and .beta..sub.d will always be equal to odd integers, and .beta..sub.d &gt;.beta..sub.n. For example, if the set of integers for .beta..sub.n and .beta..sub.d are 1 and 2, .beta..sub.n will equal 3 and .beta..sub.d will equal 5. The notation i.epsilon.I is used throughout with the same meaning and effect.
FIG. 1 describes terminal attractors with various convergence rates. Curves 1 and 2 portray attractor behavior Galedon different initial conditions. Note that the dotted curve 1' is the conventional counterpart of curve 1. Curves 3 and 4 portray the effect of attractor gain .alpha..